# Ordering is Preserved on Integers by Addition

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## Theorem

The usual ordering on the integers is preserved by the operation of addition:

- $\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$

## Proof

Recall that Integers form Ordered Integral Domain.

Then from Relation Induced by Strict Positivity Property is Compatible with Addition:

- $\forall x, y, z \in \Z: x \le y \implies \paren {x + z} \le \paren {y + z}$
- $\forall x, y, z \in \Z: x \le y \implies \paren {z + x} \le \paren {z + y}$

So:

\(\displaystyle a\) | \(\le\) | \(\displaystyle b\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a + c\) | \(\le\) | \(\displaystyle b + c\) | Relation Induced by Strict Positivity Property is Compatible with Addition |

\(\displaystyle c\) | \(\le\) | \(\displaystyle d\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle b + c\) | \(\le\) | \(\displaystyle b + d\) | Relation Induced by Strict Positivity Property is Compatible with Addition |

Finally:

\(\displaystyle a + c\) | \(\le\) | \(\displaystyle b + c\) | |||||||||||

\(\displaystyle b + c\) | \(\le\) | \(\displaystyle b + d\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a + c\) | \(\le\) | \(\displaystyle b + d\) | Definition of Ordering |

$\blacksquare$

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf Z. \, 7$